4.4 The fast variability of accreting compact objects

The strongest gravitational fields in astrophysics can be probed only with rapidly variable phenomena
around neutron stars and galactic black holes (see Figure 18 in Section 7). Such phenomena have been
discovered in almost all known accreting compact objects in the galaxy. They are quasi-periodic oscillations
(QPOs) with frequencies in the range of 1 Hz – 1 kHz that remain coherent for tens to hundreds of
cycles and follow a rich and often complicated phenomenology (for an extensive review of the observations
see [171]).

Figure 10:

The dependence of the twin QPO frequencies on the X-ray count rate observed by the
PCA instrument onboard RXTE, for the neutron-star source 4U 1820–30 [188]. The flattening of
the correlation at high frequencies has been discussed as a signature of the innermost stable circular
orbit.

4.4.1 Quasi-periodic oscillations in neutron stars

The fastest oscillations detected from accreting, weakly-magnetic neutron stars are pairs of QPOs with
variable frequencies that reach up to 1300 Hz and with frequency separations on the order of
300 Hz [171]. The origin of these oscillations is still a matter of debate. However, all current models
associate at least one of the oscillation frequencies with a characteristic dynamical frequency in a
geometrically thin accretion disk (see discussion in [124, 100, 159, 129]).

The highest dynamical frequency of a mode excited at any radius in an equatorial accretion disk around
a compact object is the one associated with the circular orbit of a test particle at that radius [8]; this is
often referred to as the azimuthal, orbital, or Keplerian frequency. A mode in the accretion
disk associated with this frequency can give rise to a long-lived quasi-periodic oscillation only
if it lives outside the innermost stable circular orbit. The azimuthal frequency at this radius
provides, therefore, an upper limit on the frequency of any observed oscillation [77, 100]. As a
result, detecting such rapid oscillations offers the possibility of measuring the location of and
understanding the properties of the region near the innermost stable circular orbit around a neutron
star.

Figure 11:

The dependence of the amplitude and quality factor of the lower kHz QPO on its
frequency for the neutron-star source 4U 1636–56 [10]. The drop of the QPO amplitude and
coherence at high frequencies have been discussed as signatures of the innermost stable circular orbit.

The signature of the ISCO on the amplitudes and characteristics of the observed oscillations is hard to
predict without a firm model for the generation of the oscillations in the X-ray flux. Two potential
signatures have been discussed, however, based on the phenomenology of the oscillations. The first one is
associated with the fact that the frequencies of the oscillations appear to increase roughly with
accretion rate. When an oscillation frequency reaches that of the innermost stable circular orbit,
one would expect its frequency to remain constant over a wide range of accretion rates [100].
Such a trend has been observed in the quasi-periodic oscillations of the globular cluster source
4U 1820–30 (See [188, 75] and Figure 10). When observations of the source obtained over
different epochs are combined, the dependence of the frequency of the fastest oscillation on
the observed accretion rate appears to flatten at a value of 1050 Hz. This is comparable
to the azimuthal frequency at the innermost stable circular orbit for a neutron
star [188].

Albeit suggestive, the interpretation of the 4U 1820–30 data relies on the assumption that the
oscillatory frequencies in an accretion disk depend monotonically on the accretion rate and, furthermore,
that the X-ray count rate is a good measure of the accretion rate. This assumption is probably justified for
short timescales (of order one day) but is known to break down on longer timescales, such as those used in
Figure 10[170]. Indeed, in a given source, the same oscillation frequencies have been observed
over a wide range of X-ray count rates and vice versa [170]. The hard X-ray color of a source,
and not the count rate, appears to be a more unique measure of the accretion rate, which is
presumably the physical parameter that determines the oscillation frequencies [95]. When the data
of 4U 1820–30 are plotted against hard color, the characteristic flattening seen in Figure 10
disappears [95].

A second signature of the innermost stable circular orbit is a potential decrease in the amplitude and
coherence of the oscillations when the region in which they are excited approaches the ISCO. Such a trend
has been observed in a number of accreting neutron stars (Figure 11 and [10, 11]) and has been
questioned on similar grounds as the study of 4U 1820–30 [94]. The most significant criticism comes
from the fact that the drop in amplitude and coherence is rather gradual and occurs over a
150 Hz range of frequencies. Even assuming that this drop is a signature of the ISCO, measuring
its location will be possible only within a detailed model of the frequencies of quasi-periodic
oscillations.

Among more model-dependent ideas, perhaps the most exciting prospect of probing strong-field gravity
effects in neutron stars with quasi-periodic oscillations comes from applying the relativistic model of
QPOs [159] to the observed correlations between various pairs of QPO frequencies [127]. In the relativistic
model, the highest-frequency QPO is identified with the azimuthal frequency of a test particle in orbit at a
given radius. The peak separation of this QPO from the second-higher frequency QPO is identified as the
radial epicyclic frequency of the test particle in the same orbit. A variant of this model can
account for the observed correlations between oscillation frequencies, when hydrodynamic effects
are taken into account [129]. Because the two observed frequencies are directly related to the
azimuthal and radial frequencies at various radii in the accretion flow, interpretation of the data
with this model can provide a direct map of the exterior spacetime of the neutron stars, to
within the 10% uncertainty introduced by the hydrodynamic corrections to the oscillation
frequencies.

4.4.2 Quasi-periodic oscillations in black holes

Pairs of rapid quasi-periodic oscillations have also been detected from a number of accreting systems that
harbor black-hole candidates [92]. The phenomenology of these oscillations is very different from the one
discussed above for accreting neutron stars. The frequencies of the rapid oscillations observed in each source
vary at most by a percent over a wide range of luminosities and their ratios are practically equal to
ratios of small integers (2:3 for XTE J1550–564 and GRO J1655–40, 3:5 for GRS 1915+105,
etc.).

The high frequencies of the oscillations observed from black-hole sources with dynamically-measured
masses demonstrate that they originate in regions very close to the black-hole horizons. In fact, requiring
the frequency of the 450 Hz oscillation observed from GRO J1655–40 to be limited by the azimuthal
frequency at the ISCO necessitates a spinning black hole with a Kerr spin parameter [160].
Moreover, the frequencies of the observed oscillations are roughly inversely proportional to the black-hole
masses, as one would expect if they were associated to dynamical frequencies near the innermost stable
circular orbit [2].

Figure 12:

(Left Panel) The intersection of the two solid lines shows the black-hole mass and spin for
the source GRO J1655–40 for which the observed 300 Hz and 450 Hz oscillations can be explained as
the lowest-order c and g-modes, respectively. The intersection of the dotted lines makes the opposite
identification of disk modes to the observed oscillatory frequencies (after [177]). (Right Panel) Each
solid line traces pairs of black-hole mass and spin for which the observed frequencies correspond to
different resonances between the Keplerian and periastron precession frequencies (after [1]). In both
panels, the horizontal dashed lines show the uncertainty in the dynamically-measured mass of the
black hole.

As in the case of neutron stars, using black-hole quasi-periodic oscillations to probe directly strong
gravitational fields is hampered by the lack of a firm understanding of the physical mechanism that is
producing them [178, 1, 135, 103]. In one interpretation, they are associated with linear oscillatory
modes that are trapped just outside the radius of the innermost stable circular orbit (for reviews
see [176, 76, 111]). The frequencies of these modes depend primarily on the mass and spin of
the black hole. Identifying the two observed oscillations with the lowest-order linear modes,
therefore, leads to two pairs of values for the mass and spin of the black hole (depending on
which oscillation is identified with which mode). For example, for the case of the black hole
GRO J1655–40, one of the inferred pairs of values agrees with the dynamically measured mass of
the black hole of and results in an estimated value of the black-hole spin of
(Figure 12 and [177]). Although compelling, this interpretation leaves to coincidence the
fact that the ratios of the oscillation frequencies are approximately equal to ratios of small
integers.

In an alternate model, the oscillations are assumed to be excited in regions of the accretion disks where
two of the dynamical frequencies are in parametric resonance, i.e., their ratios are equal to ratios of small
integers [1]. In this case, the frequencies of the oscillations depend on the mass and spin of the black hole,
as well as on the radius at which the resonance occurs. As a result, the observation of two
oscillations from any given source does not lead to a unique measurement of its mass and spin, but
rather to a family of solutions. For example, identifying the frequencies of the two oscillations
observed from GRO J1655–40 as a 3:2, a 3:1, or a 2:1 resonance between the Keplerian and the
periastron precession frequencies at any radius in the accretion disk leads to three families
of solutions, as shown in Figure 12. The dynamically-measured mass of the black hole then
picks only two of the possible families of solutions and leads to a smaller value for the inferred
spin.

Future observations of accreting neutron stars and black holes with upcoming missions that will have
fast timing capabilities, such as XEUS [185], will be able to discover a large spectrum of quasi-periodic
oscillations from each source. Such observations will constrain significantly the underlying physical model
for these oscillations, which remains the most important source of uncertainty in using fast variability
phenomena in probing strong gravitational fields.

"Probes and Tests of Strong-Field Gravity with Observations
in the Electromagnetic Spectrum"